If the equation `ax^2+by^2+cx+cy=0` represents a pair of straight lines , then

Options

A. a) `a+b=0`
B. b) `c=0`
C. c) `a+c=0`
D. d) `c(a+b)=0`

Answer

To determine the conditions under which the equation \( ax^2 + by^2 + cx + cy = 0 \) represents a pair of straight lines, we can follow these steps:

### Step 1: Write the given equation
We start with the equation:
\[ ax^2 + by^2 + cx + cy = 0 \]

### Step 2: Compare with the standard form
The standard form for the equation of a pair of straight lines is:
\[ Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0 \]
In our case, we can identify:
- \( A = a \)
- \( H = 0 \) (since there is no \( xy \) term)
- \( B = b \)
- \( G = \frac{c}{2} \)
- \( F = \frac{c}{2} \)
- \( C = 0 \)

### Step 3: Write the determinant condition
For the equation to represent a pair of straight lines, the determinant \( \Delta \) must be equal to zero. The determinant is given by:
\[ \Delta = ABC + 2FGH - AF^2 - BG^2 - CH^2 \]
Substituting the values we identified:
- \( A = a \)
- \( B = b \)
- \( C = 0 \)
- \( F = \frac{c}{2} \)
- \( G = \frac{c}{2} \)
- \( H = 0 \)

### Step 4: Substitute into the determinant formula
Substituting into the determinant formula gives:
\[ \Delta = ab(0) + 2\left(\frac{c}{2}\right)\left(\frac{c}{2}\right)(0) - a\left(\frac{c}{2}\right)^2 - b\left(\frac{c}{2}\right)^2 - 0 \]
This simplifies to:
\[ \Delta = 0 + 0 - a\left(\frac{c^2}{4}\right) - b\left(\frac{c^2}{4}\right) \]
\[ \Delta = -\frac{c^2}{4}(a + b) \]

### Step 5: Set the determinant to zero
For the equation to represent a pair of straight lines, we set \( \Delta = 0 \):
\[ -\frac{c^2}{4}(a + b) = 0 \]

### Step 6: Analyze the equation
From the equation \( -\frac{c^2}{4}(a + b) = 0 \), we can conclude that:
1. \( c^2 = 0 \) which implies \( c = 0 \)
2. \( a + b = 0 \)
3. \( c(a + b) = 0 \)

### Conclusion
Thus, the conditions for the equation \( ax^2 + by^2 + cx + cy = 0 \) to represent a pair of straight lines are:
- \( a + b = 0 \)
- \( c = 0 \)
- \( c(a + b) = 0 \)

### Final Answer
The correct options are:
- Option A: \( a + b = 0 \)
- Option B: \( c = 0 \)
- Option C: \( c(a + b) = 0 \)